General dimensional reduction of ten-dimensional supergravity and superstring

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properties are also shared by some of the models with a lower number of families. On the other hand, all these models can be obtained by a suitable truncation of the "maximal" nine-family model. A common feature of all models we will derive is their no-scale structure [6], namely the (semi-)positivity of the scalar potential and its flatness in, at least, one complex field direction(s). This includes models with extended N = 2 and N = 4 supersymmetry. Indeed, the no-scale structure is a general feature of D = 4 compactification of 10D supergravity preserving at least one supersymmetry. In what follows we will focus our discussion on the bosonic part of the lagrangian. In fact in all supersymmetric theories the fermionic part is uniquely fixed by supersymmetry, once the bosonic sector is known. In general we split the 10D indices/~=#, I ( # = l, .... 4, I = l, ..., 6) so the D = 10 bosonic supergravity fields are g~,

gul,

g,j,

Btw, 0#1, 81,1,

0,

(1)

4 December 1986

gian neglecting the contribution of the four-dimensional vector fields. After a rescaling of the D = 4 metric g~, = ~ 1/4g~4) with A= det glJ we obtain - (Ouq~)2 - ~ (Ou log A)2 + [OuglsOUgZJ ~.,~201t4"2 --

4 ~-"

_ 3A¢.2¢~/-/'2

JJl~l.I

4~

~/u~

Ia~K72 --

4 ~"

Z,ul

_1,,~,t-1/2~24 ~ ~ .' lJ-- ~e2~A-J/2H21jt¢ .

(5)

This is nothing but the scalar-Einstein sector of N : 4 supergravity coupled to 502 N---4 vector multiplets. The last two terms in eq. (5) give rise to the N = 4 scalar potential which is manifestly positive semidefinite with fiat directions being the 38 singlets under Es×E~ [eq. (1)]. To see the structure of the nonlinear a-model, we make now the following redefinitions: S = x/'~e ~ + 3ix/2D,

(6a)

where D is the dual of the B~ field [e2C'AH~p = ~u,,p,,O°D], T = Ttj =e-Ogls + ATA,~j + v / 2 B I j .

(6b)

where g~a, B~ and ~ are respectively the metric tensor, the antisymmetric tensor and the dilaton of D = 10 supergravity. The D = 10 gauge nonsinglet bosonic fields are

The complex coordinate S parametrizes the SU(I, I)/U(I ) Khhler manifold with a fixed value of the curvature, it is based on a K~ihler potential,

A~,

Js = - l o g ( S + S * ) ,

A~,

(2)

where A~' are the D = I0 gauge vector fields. The simplest dimensional reduction corresponds to a T 6 torus compactification. This demands to keep all modes given by eqs. (I), (2) (independent of the internal coordinates XI) and results in an N=4, D = 4 supergravity coupled to 502 N = 4 (vector) matter supermuitiplets with gauge symmetry U(I)6XEs ×E~. Due to the six graviphotons of the N = 4 supergravity multiplet (g#,) the overall gauge symmetry is U (1)~2× E8 × El. The scalar manifold has dimension Ds=2+6(6+n),

n=dimG=496.

(3)

The geometry of the scalar manifold is [ 7 ] SO(6, 6 + n ) SU(I, 1 ) X , SO(6) × S O ( 6 + n ) U(1)

(4)

as expected from general results on N = 4 supergravity couplings [8]. Here we only sketch the proof which leads to eq. (4). We first write the bosonic part of the 10D lagran264

(7)

and is common to all compactifications. The real coordinates TIj, A ~' parametrize the manifold SO(6, 6 + n ) / [ S O ( 6 ) × S O ( 6 + n ) ] . The geometric structure is manifest if we use projective variables P~j, P]~, such that T , j = ½[(1 + e ' )(1 - P ' ) - ' ltJ,

(8a)

Ala = [(1 - P ' ) - '

(8b)

P2],a.

Defining the rectangular matrix PIA= (PJJ, P2a) the kinetic terms of the scalar fields become OuS OuS * ( S + S , ) 2 - ½ Tr(l _ p T p ) -I OuP

X ( I - P ~ P ) - ' OuPx .

(9)

This form is SU(I, 1)XSO(6, 6 + n ) invariant. In fact by defining poj, p~a and Zo, Zt with the constraints popox

p,p,T= l

(p=po-lp,),

(10a)

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IZo12- IZt 12=1

PHYSICS LETTERSB

Z = 2S+ 1 -

.

Eq. (9) takes a manifestly SO(6, 6 + n ) x S U ( I , 1 )invariant form: Tr(D~P ° D . P ° T - D . p ' D~p 'r) + (D.Zo D"Z~o-DuZ, D,,Z~'~).

(ll)

where D. are SO (6) × U (1) covariant derivatives. It is worth mentioning that the gauging of the group U( 1) J2X G, i.e., the fact that there are six N = 4 gauge multiplets singlet under G, is essential to give rise to an N = 4 no-scale positive semidefinite potential. We will consider now consistent truncations (compactifications) of the D = I 0 theory which reduce the number of supersymmetry. The gravitinos transform as 4 under SU(4)---SO(6). So, any subgroup of SU(4) under which some gravitinos are singlets corresponds to a supersymmetric consistent truncation and, in some cases, to a real compactification. The number of singlet gravitinos corresponds to the number of unbroken supersymmetries. For instance we may obtain an N = 2 supergravity [9] decomposing SU(4)--*SU(2)' XSU(2) × U ( I ) and keeping only S U ( 2 ) ' singlets. A weaker requirement, which corresponds to an orbifold compactification on Ta/Z2XT2 is obtained by keeping only Z2 singlets of the center of SU(2)'. The latter case is actually obtained in a natural way by a two-step compactification from D = I 0 to D = 6 [10] and from D = 6 to D=4. Note that Ta/Z2, after resolution of the singularities, corresponds to K3 which is a fourdimensional Calabi-Yau manifold. Here we work out only the classification of N = 2 multiplets resulting from this compactification. To obtain a spectrum with charged matter fields we must retain singlets under S U ( 2 ) o = S U ( 2 ) @SU(2)" where SU(2)" is a subgroup of Es. Then the maximal unbroken group preserving N = 2 supersymmetry is ET)( SU (2)" the adjoint of Es decomposes as follows: 248= (133, 1 ) + (1, 3) + (56, 2 ) .

(12a)

The scalars which are in 6 of SU (4) decompose as 6 = ( 2 , 2 ) + ( 1 , 1 ) ÷ ( 1 , 1)

(12b)

4 December 1986

under S U ( 2 ) ' ® S U ( 2 ) . Therefore, the SU(2)t) singlets are 2 ( 133, 1 ) and ( 56, 2). The 2 (133, 1 ) scalars are the scalar superpartners of the E7 gauge bosons present in the N = 2 vector multiplets, while the 122 scalars transforming as (56, 2) under the ETXS(2) remaining gauge group belong to N = 2 hypermultipiers. The latter can be regarded as some coordinates of a quaternionic manifold, as dictated from N = 2 supergravity couplings. Other scalar modes are the 496 partners of the E~ gauge bosons. Notice that the 2(133, 1) scalars as well as the 496 of E~ would have been massless even if we demanded invariance under SU(2)' rather than SU(2)o. The same statement is valid for the remaining ten massless scalars coming from the SU(2)D singlets of glj, BIj, 0 and D (the dual of Bu~). Four of them come from gtj [singlets under SU(2)], four others come from B/j [a singlet and a triplet under SU(2)]. The remaining two are the 0 and D. In the absence of a charged field, the SU (2) triplet coming from BIj together with one of the seven SU(2) singlets become the scalar components of a hypermultiplet and they are coordinates of the quaternionic manifold [9] sp(2, 2 ) / [ s p ( 2 ) X s p ( 2 ) ] . The remaining six SU(2) singlets are coordinates of the reducible K~ihler manifold [ SU (1, 1 )/U (1) ] 3. They are the scalar partners of three gauge fields [ SU (2)D singlets] coming from g~a and B~a. The fourth SU(2)o singlet vector field is the graviphoton, i.e., the vector sitting in the N = 2 supergravity multiplet. lfwe consider the orbifold compactification T4/Z 2 the massless modes are actally more than those described above because one has to retain the modes which are Z2 singlets rather than the SU(2)D singlets. The additional massless modes, Zz singlets, are 3(56, 2) charged hypermultiplets, three SU(2) triplets and three SU(2) singlets which are the scalar components of three additional vector multiplets inert under ET. Finally there are six scalar fields which are the partners of the gauge fields of an additional unbroken SU (2). The extended supergravity theories so far obtained are consistent dimensional reductions (or truncations) of the 10D supergravity lagrangian. They are trivially anomaly free because of the reality properties of extended supergravities. However, as we will see below in the N = 1 case, families in complex representations of the gauge group will appear; the 265

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4 December 1986

The diagonal Z3 singlets [center of SU(3)D = S U ( 3 ) ' ~ S U ( 3 ) ] are the complex fields

Our analysis is based on 10D field theory so we are only able to give a complete description of the iagrangian which would correspond, in the string theory framework, to the massless modes of the untwisted sector. Beside this we have pointed out the need for some additional fields, not obtainable from the 10D field theory, and which are indispensable for the consistency of the model. The N = 1 supergravity lagrangian describing the effective interactions of the massless modes is entirely specified [ 11 ] in terms of the K~ihler potential J(Z, 27*) of the chiral muitiplet sector, the superpotential g(Z) (which is an analytic function of the complex scalar fields ZA) and the Yang-Mills metricf~u(Z) (analytic in the scalar fields and symmetric in the adjoint of SU (3) × E6 × E~). Actually there are only two relevant functions: G(Z, Z*) andfAswhere [ 11 ] G(Z, Z*) =J(Z, Z*) +loglg(Z) 12. In order to bring the N = 1 supergravity lagrangian [which is a truncated version of eq. (5)] in a standard supergravity form we define the complex fields T~y and S:

C7 ," with ae27 of E6 ,

.,n~- iv/-~Bij, T~j=g, Te-°+C~maC:-

(16a)

S = d e t g~ eO + 3ix/-2D.

(16b)

resulting field theory is then exposed to the risk of being anomalous. We now consider dimensional reduction with an N = 1 residual supersymmetry. To obtain the "maximal" model we pick up the smallest group under which there is only one singlet gravitino. This is the center Z3 of SU (3)' c SU (4) considered in ref. [ 5 ]. The Z3 singlet scalar fields coming from the 10D supergravity sector are

gG

'

Bij , (~, D,

(13)

where we used complex notation for the internal coordinates corresponding to the decomposition 6--,3+.~ of SU(4) into SU(3)'. The charged fields decomposition under SU(3) × E 6 c E 8 is

A ~--, C[ ~'~) + C~ 3'27) "-{-C~1'78) +C~8'l) + (h.c.) , a ~ E 8 .

n e 3 o f S U ( 3 ) , ie3 o f S U ( 3 ) ' .

(14)

(15)

The surviving N = 1 gauge fields '~ - u".78) , '~ - u~8.t ) are in the adjoint of the SU(3) × E 6 gauge group [4]. The present model contains nine 27 of families which are triplets under the horizontal gauge SU(3) symmetry. The gravity sector corresponds to 10 chiral multiplets which are singlets under the surviving SU (3) X E6 × E~ gauge group. This model, containing chiral fermions in the 3 of the SU(3) gauge group, is anomalous. The simplest way to cancel this anomaly is to have 81 additional chiral multiplets in the :] of SU(3). However, from the 10D field theory the :~ chiral fermions are also in the 2-7 of E 6. It is therefore impossible to obtain an anomaly-free model with a net number of chiral generations. A way out of this problem is to eliminate the SU(3) vector multiplets, which is a totally arbitrary procedure. On the other hand, the additional chiral fermions needed to cancel the anomaly come naturally in the orbifold (T6/Z 3) string compactification. They arise from the "twisted" sectors of the theory and they are in the (3, 1 ) of SU (3) X E6 [ 4 ]. There are precisely 27 copies of them. The remaining additional modes are singlets under SU(3) transforming as 27 under E 6. There are equally 27 copies of these extra families [4]. 266

Then, the N = 1 K~ihler potential is, Z = ( S , T,j,

c.o),

J( Z, Z * ) = - l o g ( S + S * ) - l o g det( Tij + T~7 - 2C,"~ C~/~a) ,

(17)

the superpotential is

g( Z ) =d,~bc~,,,,l~°k "~''~ p"b P tc

(18)

where e,,,~ is the antisymmetric tensor of SU (3) and dab,. the symmetric tensor of the 27 of E6; the Yang-MillsfAn(Z) function is

fas(Z)=S6~o,

A, BeSU(3)×E6×E'8.

(19)

The "maximal" nine-family K~ihler manifold defined from eq. (17) is SU(I, - - ×1) U(I) n=27.

SU(3,3+ 3n) SU(3) x S U ( 3 + 3 n ) × U ( I ) ' (20)

Obviously this manifold is a truncation of the N = 4 model given in eq. (4). Note that the geometry of the K~hler manifold eq. (17) quarantees the positive semidefiniteness of the scalar potential irrespec-

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PHYSICS LETTERS B

tively of the form of the superpotential g ( Z ) . This grassmannian manifold shows in particular that the CP ~ SU(n,1)/[SU(n) x U ( 1 ) ] spaces previously considered [6] are special examples of the geometries of no-scale models. The complete I~hler manifold of the 4D effective action, corresponding to the orbifold compactified string, must then be a I~hler space which contains that of eq. (20) as a proper submanifold. Let us denote by X~'F) and YT(r) the additional modes coming from the "twisted" sectors ( F = 1, 2,..., 27, ae 27 of E6, and r/(and 7) e3 o f S U ( 3 ) [ S U ( 3 ) ' ] ) , the G function of the complete theory must obviously contain these extra modes. The Yi"tr~ fields can appear in the G function in a way very similar to the C~,~ fields. A simple possible form for G when X~.) = 0 (which is a maximally symmetric one) is then G=-

log(S+S*)

(21)

where the superpotential is now given by g

.

r,

= t m n l [ t:

ilk A t'~ma ¢ " n b t"l,-" °tabc~'.~i ~'-~] "-'k

i k v~r~ Y,F,) m n ~)y I k ( i . . ) ] , +2Y., Y~(+'

T h e Z 6 invariance keeps only singlets and triplets under S U ( 2 ) o c S U ( 3 ) n . The gauge group in this case becomes U(1 ) x S U ( 2 ) x E 6 x E ~ . The model contains five families, five singlets and the universal S singlet. The five family Kiihler potential is

J c s ) ( Z , Z*) = - l o g ( S + S*)

- log det( wu + w~,j- 2c,m~ C~'a) (23)

-Iog(x + x*- 2C~C~) ,

where now w0 is a 2X2 hermitian matrix and m e 2 ofSU(2)o, x is a complex field, and ae 27 of E6. The corresponding superpotential is g( 5) ( Z ) = dah,.~l,,, ~ 'J C ~ Cf't' C~ ,

(24)

where now et,, is the antisymmetric tensor of SU(2)o. ThefA~(Z) function is model independent [see eq. (19)]. The geometry of the scalar manifold is SU(I, 1) SU(2,2 +2n) X U(I) SU(2) x S U ( 2 + 2 n ) x U ( I )

- l o g d e t ( i/',i + T~,, - 2C7~C~j~ ,7 ) +loglgl 2 - 2 ~27 Y~,~'v)Yj(F) /

4 December 1986

(22)

with 2ykr~F~ constant coefficients, if we restrict ourselves to a trilinear superpotential. (In general, when X~F) ¢0, additional terms of the form Tr X 3 are present in the superpotential g.) The X dependence of G, however, is much harder to guess without explicit string calculations.The only restriction on the form of G comes from the requirement of semi-positive definiteness of the scalar potential and of the flatness in the Tdirections which, however, is very mild. N= 1 supersymmetric models with lower number of families and Tsinglets are obtained by demanding the massless modes to be singlets under larger symmetry than Z3. Needless to say, they must all be thought as consistently embedded in string theories on orbifolds, and describing its untwisted sectors. We can consider for instance singlets under Z6 discrete group which is generated by the 3 X3 matrix M = d i a g ( - - e 2in/3, - - e 2in/3, e2in/3), notice that M 6 = 1.

x

SU(1, 1 + n ) SU(I+n)×U(1) '

n=27.

(25)

Another example, leading to a three-family model and U(I ) × U ( I ) × E 6 x E~as gauge group is obtained by retaining singlets under Z~2. Z~2 is the discrete group which is generated by the 3X3 matrix M=diag(ie 2i~/3, - i e 2i~/3, e2i~/3)(MI2= 1). In this case the K~hler manifold is J ( 3 ) ( Z , Z * ) = - log(S+S*) 3

- ~ log(T, + ~ - 2 C ~ C ~ )

(26)

:=1

and the superpotential g ( 3 ) ( Z ) --d~b~C al C2C~33 b •

(27)

The unbroken gauge group is U(1 ) x U ( 1 ) XE6XE~ and the Kiihler manifold is

SU(I,U(I~I)XkSU(I ( SU(I, l+n) )~ . +n)xU(1 )

(28)

Note that under the two U (1) the three families have charges (I,1), - 1,1 ), ( 0 , - 2 ) , respectively. The five- and three-family models are also related to orbifold compactifications, namely on T6/Z6 and T6/Z~ 2, respectively. 267

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Examples with two and one families can also be obtained by retaining only singlets u n d e r S U ( 2 ) × U ( 1 ) or u n d e r the full S U ( 3 ) o , respectively. The one-family " m i n i m a l " model was considered in ref. [5]. It has a CP ~ K~ihler geometry and gauge group E 6 X E~. The K~ihler potential of the twofamily model is

J~2)(Z, Z * ) = - l o g ( S + S*)

- l o g ( x + x * - 2 C ~ C - ~ ~) ,

(29)

and the superpotential is g~2)( Z) __ a b c . -d~bcCwC~Cx

(30)

The Kahler manifold is

SU(l,l+n) ~2 SU(I, 1) X

U-(-i)

"

The remaining gauge group is U (1) X E6 × E~; Cw and C~ have + 1 and - 2 U ( I ) charges, respectively. Even in this case the ( K a l u z a - K l e i n ) U (1) gauge grOUp is anomalous. In this work we have derived various models coming from dimensional reduction of 10D supergravity. We have shown that those models which correspond to orbifold compactifications give rise to the complete low-energy lagrangian for the "untwisted" sector of the corresponding string theory. In this approach in particular, all the nonpolynomial a-model interactions are taken into account. It is worth stressing that from our analysis we were able to find another motivation twin of the request of modular invariance in string theory on orbifolds [4,12] for the presence of "twisted" sectors: the anomaly cancelation in the four-dimensional massless sector. Also, we get some restrictions on the possible form for the lagrangian of the "twisted" sectors.

268"

One of us (C.K.) would like to thank the Physics Department of UCLA for his hospitality. M.P. would like to thank LBL and UCB where a part of the work was done. We are grateful to Sharam Hamidi for useful comments and discussions about some properties of the models presented here and their relations with string on orbifolds.

References

-- 2log(w+ w*-- 2ca.C*~ ~)

S(I+n)xU(1)}

4 December 1986

[ I ] F. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Nucl. Phys. B 195 (1982) 97; G. Chapline and N. Manton, Phys. Lett. B 120 (1983) 105. [2] M.B. Green and J.H. Schwartz, Phys. Lett. B 149 (1984) 117; D.J. Gross, J.A. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Nucl. Phys. B 260 (1985) 569. [3] P. Candelas, G.T. Horowitz, A. Strominger and E. Winen, Nucl. Phys. B 258 (1985) 46. [4] L. Dixon, J.A. Harvey, C. Vafa and E. Wittcn, Nucl. Phys. B 261 (1985) 651; String on orbifolds II, Princeton preprint (1985). [5] E. Witten, Phys. Lett. B 155 (1985) 151. [61 E. Crammer, S. Ferrara, C. Kounnas and D.V.Nanopoulos. Phys. Lett. B 133 (1983) 61; J. Ellis, A.B. Lahanas, D.V. Nanopoulos and K. Tamvakis, Phys. Left. B 134 (1984) 42; J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B 241 (1984) 406; Nucl. Phys. B 247 (1985) 373. [7]J.-P. Deringer and S. Ferrara, in: Supersymmetry and supergravity '84, eds. B. de Wit, P. Fayet and P. van Nieuwenhuizen (World Scientific,Singapore, 1984) p. 159. [8] M. de Roo, Phys. Lett. B 156 (1985) 331; Nucl. Phys. B 255 (1985) 515; E. Bergshoeff. G.T. Koh and E. Sezgin, Nucl. Phys. B 245 (1984) 89. [9] B. de Wit and A. van Proeyen, Nucl. Phys. B 245 (1984) 89. [10] M.B. Green, J.H. Schwarz and P.C. West, Anomaly-free chiral theories in six dimensions, CALT-68-1210. [ 11 ] E. Cremmer, S. Ferrara, L. Girardello and A. van Proeyen, Phys. Lett. B 116 (1982) 231; Nucl. Phys. B 212 (1983) 413. [ 121 C. Vafa, Harvard preprint HUTP-86/A011 (1986).